If \(f^3=1\)  and all minimal relations are quaternion, then the algebra \(A\)  is a weighted surface algebra (as in [1]}, [2]}, [3]}).
When \(f^3=1\) , and some but not all minimal relations are biserial, we get algebras generalizing
algebras of semidihedral type (which occur when \(Q\)  has at most three vertices), as in [4]}. The condition \(f^3=1\)  has
geometric origin, it is closely related to surface triangulation.
